Finite cyclic groups, subgroups of order 5, and normalizers

[cauchys_pet]

hey! great to find such an informative website...
i'm an undergrad math student and have lots of problems with group theory.i hope you'll all help me enjoy group theory...
my teacher put forward these question last week and I've been breaking my head over them without much success :
1. let G be a finite cyclic group of order p^n, p being prime and n >=0. if H and K are subgroups of G then show that either H contains K or K contains H.
i started out supposing the contrary but i wonder if I'm on the right track. i don't think it helps.

2.if G is a group of order 30 show that G has atmost 7 distinct subgroups of order 5.
can i say this : let H be a subgroup of order 5 then the number of distinct left cosets of H in G is 6. so are we done?!

3.let G be a group such that intesection of all subgroups of G different from {e}. then prove that every element of G has finite order.

4. give an example to show that a subgroup of index 3 may not be a normal subgroup of G.
thanks again for the help.

 

[matt grime]

1. G is cyclic what does that mean? so what can you say about the elements in H and K in terms of this?


2. Is nothing to do with cosets. Suppose H and K are subgroups of order 5, then HnK is a subgroup whose order divides 5, so it follows HnK=H=K or HnK={e} So the subgroups are either equal or contain only one element in common. So if there 7 (or fewer) distinct subgroups of order 5 these contain 7*4+1=29 disticnt elements: they all contain e, and 4 other elements each that appear in exactly one subgroup. If there were more than 7 then what would happen?

3 makes no sense.

4. Hmm, can you think of any small subgroups that have a subgroup of index 3 that aren't abelian? Try the smallest such (it has order 6...)

 

[mansi]

have been thinking about prob 3...i guess you take A6 (the set of even permutations of 6 elements), possibly find a subgroup H of order 4 and then look at the 3 distinct cosets of H.

 

[cauchys_pet]

sorry, i goofed up problem number 2
it says, let G be a group such that intersection of all subgroups of G different from {e} is not {e}. prove that every element is of finite order.

 

[matt grime]

suppose there is an element of infinite order, g say.

For all r in N let C_r be the cyclic group generated by g^r,

What is the intersection of all these?